Albertson–Grossman–Haas partial list coloring conjecture

Let $G$ be an $n$-vertex graph with list-chromatic number $\chi_{\ell}$. For a positive integer $t\leq\chi_{\ell}$, assign each vertex a list of $t$ colors, and let $\lambda_t$ be the minimum, over all such list assignments, of the maximum number of vertices that can be properly colored from their lists. Albertson–Grossman–Haas conjecture. If $t\leq\chi_{\ell}$, then

$$\lambda_t\geq\frac{tn}{\chi_{\ell}}.$$

Equivalently, at least $tn/\chi_{\ell}$ vertices can be colored from every assignment of lists of size $t$. The conjecture is known in the cases $t=1$ and $t=\chi_{\ell}$, but remains open for $1<t<\chi_{\ell}$.